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4 THE PRINCIPLE OF ANALYTIC CONTINUATION 207
4. Let (pn zn), (gn zn) be two power series with complex coefficients, and radius of conver-
gence 7^0, and let f(z) =^pnz", g(z) =]>] qn z" in a neighborhood U of 0 where both |
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series are absolutely convergent. Suppose q0 =#(0) ^ 0; then there is a power series
£ cnzn which is absolutely convergent in a neighborhood V <=• U of 0 and has a sum
R
equal to/(z)/#(z) in V (remark that the series (zn) is convergent for \z\ < 1, and use
(9.2.2)). If all the qn are >0, the sequence (qn+ilqn) is increasing, the pn are real and such that the sequence (pnlqn) is decreasing (resp. decreasing), show that cn ^ 0 (resp. cn ^ 0) for every n>l. (Write the difference
Pn Pn-l
qn qn-i
as an expression in the qk and ck , and use induction on «.) Deduce from that result
that all the derivatives of jc/log(l — x) are <0 for 0 < x < 1.
5. Let gk (1 ^ k ^ p) be p scalar entire functions defined in K*. If /is an entire function
defined in Kp, then/(#i, . . . , gp) is an entire function in Kq.
6. Let P be the disk \z\ ^ r in C. Show that
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55.'
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n dx dy = 0 for any integer n > 1.
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(Use the fact that Lebesgue measure on R2 is invariant by rotation (cf. (14.3.9)).)
Deduce from that relation that for every function / which is analytic in an open set A c C containing P, one has |
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55.
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Show similarly that
fr znzmdxdy=Q if m •-
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rr ^^+2
JOl-^-TM
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4. THE PRINCIPLE OF ANALYTIC CONTINUATION
(9.4.1) In Kp, let P, Q be two open polydisks of centers a, b, such that
P n Q ^ 0. Let (cni mm,np(x^ — a^f1 ...(xp — ap)np) be a power series in the xt — aiy absolutely summable in P, and let f(x) be its sum. Let (dni... np(*i — *i)ni - - • (xp - bp)np) be a power series in the Xi — bi9 absolutely summable in Q, and let g(x) be its sum. If there is a nonempty open subset U of P n Q such that f(x) = g(x) for any x e U, then f(x) = g(x) for any x e P n Q.
Let u e U, and let v be any point of P n Q; then the segment joining u and v
is contained in P n Q by (9.1.1). Let h(t) =f(u + t(v - u)) - g(u + t(v - u)) with t real; by (9.3.2), this is an analytic function of / in an open interval I |
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